Arguments

x

vector of quantiles.

q

vector of quantiles.

p

vector of probabilities between 0 and 1.

n

sample size. If length(n) is larger than 1, then length(n)
random values are returned.

meanlog

vector of means of the normal (Gaussian) part of the distribution on the
log scale. The default is meanlog=0.

sdlog

vector of (positive) standard deviations of the normal (Gaussian)
part of the distribution on the log scale. The default is sdlog=1.

p.zero

vector of probabilities between 0 and 1 indicating the probability the random
variable equals 0. For rzmlnorm this must be a single, non-missing number.

Details

The zero-modified lognormal (delta) distribution is the mixture of a
lognormal distribution with a positive probability mass at 0. This distribution
was introduced without a name by Aitchison (1955), and the name
Δ-distribution was coined by Aitchison and Brown (1957, p.95).
It is a special case of a “zero-modified” distribution
(see Johnson et al., 1992, p. 312).

h(y; μ, σ, p) =

p

for y = 0

(1 - p) f(y; μ, σ)

for y > 0

Note that μ is not the mean of the zero-modified lognormal
distribution on the log scale; it is the mean of the lognormal part of the
distribution on the log scale. Similarly, σ is
not the standard deviation of the zero-modified lognormal distribution
on the log scale; it is the standard deviation of the lognormal part of the
distribution on the log scale.

Note that when p.zero=p=0, the zero-modified lognormal
distribution simplifies to the lognormal distribution.

Value

Note

The zero-modified lognormal (delta) distribution is sometimes used to
model chemical concentrations for which some observations are reported as
“Below Detection Limit” (the nondetects are assumed equal to 0).
See, for example, Gilliom and Helsel (1986), Owen and DeRouen (1980), and
Gibbons et al. (2009, Chapter 12). USEPA (2009, Chapter 15) recommends this
strategy only in specific situations, and Helsel (2012, Chapter 1) strongly
discourages this approach to dealing with non-detects.

A variation of the zero-modified lognormal (delta) distribution is the
zero-modified normal distribution, in which a
normal distribution is mixed with a positive probability mass at 0.

One way to try to assess whether a zero-modified lognormal (delta),
zero-modified normal, censored normal, or censored lognormal is the best
model for the data is to construct both censored and detects-only probability
plots (see qqPlotCensored).

Owen, W., and T. DeRouen. (1980). Estimation of the Mean for Lognormal Data
Containing Zeros and Left-Censored Values, with Applications to the Measurement
of Worker Exposure to Air Contaminants. Biometrics36, 707-719.